Parallel mechanism based automated fiber placement system

ABSTRACT

The present invention introduces a new concept of applying a parallel mechanism in automated fiber placement for aerospace part manufacturing. The proposed system requirements are 4DOF parallel mechanism consisting of two RPS and two UPS limbs with two rotational and two translational motions. Both inverse and forward kinematics models are obtained and solved analytically. Based on the overall Jacobian matrix in screw theory, singularity loci are presented and the singularity-free workspace is correspondingly illustrated. To maximize the singularity-free workspace, locations of the two UPS limbs with the platform and base sizes are used in the optimization which gives a new design of a 4DOF parallel mechanism. A dimensionless Jacobian matrix is also defined and its condition number is used for optimizing the kinematics performance in the optimization process. A numerical example is presented with physical constraint considerations of a test bed design for automated fiber placement.

FIELD OF THE INVENTION

The present invention generally relates to parallel mechanism inautomated fiber placement for aerospace part manufacturing.

BACKGROUND OF THE INVENTION

Automated fiber placement (AFP) is an important manufacturing process incomposite aerospace part manufacturing and has attracted much interestsince future aircraft programs, such as the Boeing 787 and AirbusA350XWB, contain more than 50% by weight of advanced compositecomponents. Also, the use of robot manipulators increases theflexibility of the fiber placement process and allows for thefabrication of more complex structures. Existing AFP research discussesproductivity, steering and control, processing conditions, materials,layup modeling and simulation, and functional integration. Robotics workmainly focuses on path planning for AFP while typically a point-cloud isgenerated for the AFP head to follow to lay the material onto the mold.

Although many robot based AFP systems have been proposed and studied,this technology is still not widely used in industry where manual lay-upis still the main method due to cost constraints and level of complexityof molds. Furthermore, all previous work used serial robots as operationarms to hold the fiber placement head due to the fact that they arewidely developed and used in automatic industry. However, serial robotsgenerally have low stiffness and large inertia due to their seriallyconnected structure, which affects their force and precision performancein high-compact-force applications, like AFP. In contrast to serialrobots, parallel robots have multiple support limbs with low inertia,high structure stiffness, good positioning accuracy and high speeds.Based on this, they are widely used in the industrial applicationsrequiring high speed and stiffness. Thus in this paper for the firsttime, a parallel mechanism is introduced in AFP and an optimal design isproposed as a basis for AFP.

In general a 6DOF platform is needed for the AFP operation to haveflexibility in manufacturing all kinds of parts with complex moldsurfaces. Considering the need of a moving platform to support theparallel mechanism and spindle rotation of the placement head on theparallel mechanism, a 4DOF parallel mechanism with 2T2R (twotranslations and two rotations) will be sufficient for automated fiberplacement. In parallel mechanism research, 6DOF ones have been studiedextensively with the Stewart-Gough platform with later focus moving toparallel mechanisms with less than 6DOF represented by many 3DOF ones.Due to the complexity of coupling between rotation and translation andsingularity issues, 4DOF parallel mechanisms have not been investigatedmuch and related work is mainly on synthesis. A class of asymmetrical2T2R parallel mechanisms was synthesized in while symmetrical ones wereobtained in using screw theory. Focusing on two rotation motions, andsynthesized new 2T2R parallel mechanisms using general function set andLie group theory respectively. Considering the requirement of a movingbase (one translation) and a rotating spindle (one rotation), 2T2Rparallel platforms have been used in 5- or 6-axis machine tools. A2PRR-2PUS parallel mechanism with a moving platform formed by two partsjoined with a revolute joint was proposed for a 5-axis machine tool.Reachable workspace of a 2PSS-2PUS parallel mechanism with two sphericaljoints coincided was studied in for machine tool applications while fourpossible singularity configurations of a 2UPR-2UPS parallel mechanismwere obtained in. Recently, for needle manipulation tasks a class of2T2R parallel mechanisms were synthesized using screw theory. However,little work has been found on singularity-free workspace analysis andoptimal design of 2T2R parallel mechanisms.

SUMMARY OF THE INVENTION

Therefore, the object of the present invention is a 2T2R parallelmechanism is for automated fiber placement where the motion is realizedby a 2RPS-2UPS topology. The mechanism size is mainly determined by thetwo RPS (revolute-prismatic-spherical joints) limbs. Finding theoptimized locations of the two UPS (universal-prismatic-sphericaljoints) limbs is the main objective in the optimal design for giving alarge singularity-free workspace and good kinematics performancerepresented by the condition number of a dimensionless Jacobian matrix.To compare the workspace volume for translation and rotation, anangle-based 3D space is proposed to have a uniform unit by representingthe translations in rotations. An optimized mechanism configuration isfound and the effect of joint components is also demonstrated, resultingin a practical design for AFP and other applications.

Although many robot based Automated Fiber Placement (AFP) systems havebeen proposed and used in composites manufacturing industry, butparallel robots are still not used in composites manufacturing.Furthermore, all the previous work applied serial robots as operationarms to hold the fiber placement head due to the fact that they arewidely developed and used in automatic industry. However, serial robotsgenerally have low stiffness and large inertia due to their seriallyconnected structure, which affects their force and precision performancein high-compact-force applications, like AFP. In contrast to serialrobots, parallel robots may decrease the weight of lay-up heads, offermultiple support limbs with low inertia, high structure stiffness, goodpositioning accuracy and high speeds.

The proposed 2T2R (two translation and two rotation) parallel mechanismconsists of two RPS (revolute joint, prismatic joint and sphericaljoint) limbs and two UPS (universal joint, prismatic joint and sphericaljoint) limbs. In each limb, the revolute or universal joint is attachedto the base while the spherical joint is on the platform, and prismaticjoints are connected in between. The two revolute joints in the RPSlimbs are located parallel to each other on the base and make the twolimbs work in the same plane perpendicular to these revolute joints.Joints on the platform or base are in 3D space and not constrained onthe same plane (the example in the paper made all joints on the plane).All the joints in the system can be commercialized ones or customerdesigned. For example, the prismatic joint can be electric, hydraulic,pneumatic, or any other forms that can provide linear motion with enoughpower. The spherical joints can be ball joint or serially connectedrevolute joints and the universal joint can be cross-link connected orserially connected revolute joints.

The 4-DOF Parallel Mechanism can be combined with a moving base toincrease the workspace and add one rotational motion to AFP head on themoving platform to form a 6-DOF system for automated fiber placementmanufacturing process. The system can be used to lay-up very complexparts with the ability to adjust to any kind of reinforcing fibers andmatrix materials such as thermoset and thermoplastic tapes/prepregs anddry carbon fiber unidirectional materials.

In addition to AFP manufacturing, the proposed mechanism can also beused in many applications with motion operation, like cameraorientation, material grasping and manipulation, machining tool, lasercutting, 3D printing, stabilization platform, motion simulator,automatic painting, automatic welding, Non Destructive Testing (NDT)etc.

As a first aspect of the invention, there is provided a robot apparatuscomprising:

-   -   a parallel mechanism comprising two Revolute-Prismatic-Spherical        joint (RPS) limbs and two Universal-Prismatic-Spherical joint        (UPS) limbs.

Preferably, the parallel mechanism is adapted to be used for compositesmanufacturing.

Preferably, the parallel mechanism is adapted to operate an AutomatedFiber Placement Head (AFP).

Preferably, the parallel mechanism provides a 4 Degree-Of-Freedom (DOF)movement comprising two rotations and two translations.

Preferably, the prismatic joints are adapted to provide linear motionwith enough power.

Preferably, the prismatic joints are electric, hydraulic or pneumatic.

Preferably, the spherical joints are ball joint or serially connectedrevolute joints.

Preferably, the universal joints are cross-link connected or seriallyconnected revolute joints.

Preferably, the robot further comprises

-   -   a platform; and    -   a base;        wherein the revolute and universal joints are adapted to be        connected to the base, the spherical joints are adapted to be        connected to the platform and the prismatic joints are adapted        to be connected intermediate the revolute/universal and the        spherical joints.

Preferably, the two revolute joints in the RPS limbs are locatedparallel to each other on to the base and adapted to make the two limbswork in the same plane perpendicular to the revolute joints.

Preferably, the joints are in 3D space and not constrained on the sameplane.

Preferably, the base is movable enabling a 6 degree-of-freedom movementfor the parallel mechanism.

Preferably, the robot apparatus further comprises an Automated FiberPlacement Head (AFP) adapted to be connected to the base.

Preferably, the robot apparatus is adapted for use in compositesmanufacturing.

Preferably, the robot apparatus is adapted to reinforce fibers andmatrix materials comprising thermoset and thermoplastic tapes orprepregs and dry carbon fiber unidirectional materials.

Preferably, the robot apparatus is adapted for use in cameraorientation, material grasping and manipulation, machining tool, lasercutting, 3D printing, stabilization platform, motion simulator,automatic painting, automatic welding or non destructive testing.

BRIEF DESCRIPTION OF THE DRAWINGS

The advantages and features of the present invention will become betterunderstood with reference to the following detailed description taken inconjunction with the accompanying drawings, in which:

FIG. 1 illustrates a serial robot arm based AFP system according to oneembodiment of the present invention;

FIG. 2(a) illustrates a 2RPS-2UPS parallel mechanism based AFP systemaccording to one embodiment of the present invention;

FIG. 2(b) illustrates a 2RPS -2UPS parallel mechanism for a 2RPS-2UPSparallel mechanism based AFP system according to one embodiment of thepresent invention;

FIG. 3 illustrates a general 2RPS-2UPS parallel mechanism and itsrepresentative kinematics model according to one embodiment of thepresent invention;

FIG. 4 illustrates selected points and directions on the platformaccording to one embodiment of the present invention;

FIG. 5(a) illustrates singularity loci and first singular configurationaccording to one embodiment of the present invention;

FIG. 5(b) illustrates singularity loci and second singular configurationaccording to one embodiment of the present invention;

FIG. 6 illustrates maximum singularity-free workspace according to oneembodiment of the present invention;

FIG. 7 illustrates effect of parameter a₁ and b₁ according to oneembodiment of the present invention;

FIG. 8 illustrates effect of parameter φ_(b1) and φ_(a1) according toone embodiment of the present invention;

FIG. 9 illustrates effect of parameters φ_(a1) and φ_(a3) according toone embodiment of the present invention;

FIG. 10 illustrates effect of parameters φ_(b1) and φ_(b3) according toone embodiment of the present invention;

FIG. 11 illustrates the optimized mechanism configuration according toone embodiment of the present invention;

FIG. 12 illustrates effect of parameters λ_(l) and ψ_(max) according toone embodiment of the present invention;

DETAILED DESCRIPTION OF THE INVENTION

2. A 2T2R Parallel Mechanism Based AFP System

In AFP systems, fiber tows are guided by a fiber-processing headattached to the end-effector of a robot and carefully placed following apre-defined robot trajectory as in FIG. 1. To tack the tows on the mold,they are heated and compacted at the same time when the robot is moving.Since the ratio of the mass of payload over the mass of the robot istypically small for 6DOF serial robots, like the KUKA robot in FIG. 1,the ratio is about 210 kg/1150 kg=0.18, the ratio can be larger than 10for parallel robots. In this case, for the same payload of 210 kg, a 21kg parallel robot might be enough to support it. Following this, theadvantage is obvious considering the space occupation and system cost.Although parallel robots suffers from the smaller workspace than serialones, a moving base is generally used to enlarge the workspace and it isalso common for the serial robots in the industry, like the one in FIG.1.

Considering one translation DOF giving by a moving base to enlarge theworkspace and one rotation DOF from AFP head on the moving platform, aparallel mechanism with two translational and two rotational DOFs issufficient to avoid redundant movement. The proposed AFP system is shownin FIG. 2(a) which includes a rail-based moving base. The moving basemoves along the x-axis and the AFP head can rotate along its axis whichis perpendicular to the parallel mechanism platform. Thus, the parallelmechanism needs to have two translations (2T) along y-axis and z-axisand two rotations (2R) about x-axis and y-axis.

The proposed 2T2R parallel mechanism consists of two RPS limbs and twoUPS limbs as shown in FIG. 2(b). In each limb, the revolute or universaljoint is attached to the base while the spherical joint is on theplatform, and prismatic joints are connected in between. The tworevolute joints in the RPS limbs are located parallel to each other onthe base and make the two limbs work in the same plane perpendicular tothese revolute joints. It is constrained that all the joint centers onthe base or platform are in the same plane. Since the UPS limbs do notconstrain the platform and each RPS limb provides one constraint forcepassing through the spherical joint center and parallel to the revolutejoint, the platform is subjected to two parallel constraint forcesduring all the motion and lose a translation motion along these forcesand a rotation motion perpendicular to the plane containing these twoconstraint forces. The four prismatic joints are selected as theactuation for the 4DOF parallel mechanism.

The initial assumption of this study is that the two RPS limbs arerelatively fixed while the two UPS limbs can be freely chosen. This willresult in variable configurations with different singularity-freeworkspace and kinematics performance. Thus the following study shows amethod of optimizing locations of the two UPS limbs with respect to theRPS limbs and the sizes of the platform and base, to get maximumsingularity-free workspace with acceptable kinematics properties. FIG. 2shows a general configuration where the two UPS limbs are on eithersides of the RPS limbs, but the optimal result later has the two UPSlimbs on the same side to give a larger singularity-free workspace.

3. The 2T2R Parallel Mechanism and Analytical Inverse and ForwardKinematics Analysis:

3.1 Variable Configurations of the 2RPS-2UPS Parallel Mechanism and itsKinematics Model:

A representative kinematics model of the 2RPS-2UPS parallel mechanism isshown in FIG. 3 where the four limbs are numbered from 1 to 4 and thetwo RPS limbs are number 2 and number 4 respectively. Let B_(i) denotethe center of the base joint and A_(i) denote the center of the platformspherical joint of the ith (i=1, 2, 3, 4) limb. Set a base coordinateframe oxyz at the middle point o of B₂B₄ with z-axis perpendicular tothe base plane formed by B₁B₂B₃B₄ and x-axis perpendicular to the lineB₂B₄. Then y-axis is in line with oB₂ as in FIG. 3. Attach a platformcoordinate frame o′x′y′z′ at middle point o′ of A₂A₄ with z′-axisperpendicular to the platform plane and x-axis perpendicular to A₂A₄.When the platform is at the initial configuration, the platformcoordinate frame is parallel with the base coordinate frame. Based onthese coordinate system settings, the platform will have two translationmotions along y-axis and z-axis with two rotations about x-axis andy-axis.

Let a_(i) denote the constant position vector of platform joint centerA_(i) in the platform coordinate frame o′x′y′z′ and b_(i) be theconstant vector of base joint center B_(i) expressed in the basecoordinate frame oxyz. Then the limb distance constraints can bedescribed as

$\begin{matrix}\{ {\begin{matrix}{{( {{R \cdot a_{i}} + p - b_{i}} )^{T}( {{R \cdot a_{i}} + p - b_{i}} )} = l_{i}^{2}} \\{a_{i} = {a_{i}( {{\cos ( \varphi_{ai} )},{\sin ( \varphi_{ai} )},0} )}^{T}} \\{b_{i} = {b_{i}( {{\cos ( \varphi_{bi} )},{\sin ( \varphi_{bi} )},0} )}^{T}}\end{matrix}( {{i = 1},2,3,4} )}  & (1)\end{matrix}$

where l_(i) is the length of limb i, R is the 3 by 3 rotation matrixcovering two rotations about x-axis and y-axis, p=(0,p_(y),p_(z))^(T) isthe translation vector of point o′ in the base coordinate system oxyz,a_(i) is the distance from point A_(i) to o′ and φ_(ai) is its angle inthe platform coordinate frame measured from x′-axis, b_(i) is thedistance from point B_(i) to o and φ_(bi) is its angle in the basecoordinate frame measured from x-axis. Based on the configuration,φ_(b2)=φ_(a2)=π/2, φ_(b4)=φ_(a4)=3π/2, a₂=a₄=a, b₂=b₄=b, a₁=a₃, b₁=b₃will be used in the optimization design.

Equation (1) gives the general geometric constraint of the 2RPS-2UPSparallel mechanism. It is noted that given R and p the inverse kinematicsolution can be obtained directly from (1) to give the four input limblengths l_(i). The forward kinematics analysis in general is morecomplex and the following Section shows an analytical solution.

3.2 Analytical Forward Kinematics:

-   Based on the geometric structure of the mechanism in FIG. 3, the    vector of the spherical joint center A_(i) in the base coordinate    frame oxyz, a_(io), is given by:

$\begin{matrix}\{ \begin{matrix}{a_{2o} = {b_{2} + ( {0,{{- l_{2}}{\cos ( \alpha_{2} )}},{l_{2}{\sin ( \alpha_{2} )}}} )^{T}}} \\{a_{4o} = {b_{4} + ( {0,{l_{4}{\cos ( \alpha_{4} )}},{l_{4}{\sin ( \alpha_{4} )}}} )^{T}}} \\{a_{1o} = ( {x_{1},y_{1},z_{1}} )^{T}} \\{a_{3o} = {{k_{1}a_{1o}} + {k_{2}a_{2o}} + {k_{4}a_{4o}}}}\end{matrix}  & (2)\end{matrix}$

where a_(i) (i=2,4) is the angle between limb i and line B₂B₄ as in FIG.2(b), and k_(i) (i=1,2,4) are constant coefficients to linearly expresspoint A₃ by the other three points in the platform plane.

Then considering the geometric shape of the platform and limb lengths,the following equations exist:

$\begin{matrix}\{ \begin{matrix}{( {\sqrt{2}a} )^{2} = ( {a_{2o} - a_{4o}} )^{2}} \\{{2{aa}_{1}{\cos ( \varphi_{a\; 1} )}} = {( {a_{2o} - a_{4o}} ) \cdot ( {a_{1o} - {( {a_{2o} + a_{4o}} )/2}} )}} \\{a_{1}^{2} = ( {a_{1o} - {( {a_{2o} + a_{4o}} )/2}} )^{2}} \\{l_{1}^{2} = ( {b_{1} - a_{1o}} )^{2}} \\{l_{3}^{2} = ( {b_{3} - a_{3o}} )^{2}}\end{matrix}  & (3)\end{matrix}$

where the first one represents the distance between spherical joint A₂and A₄, the second one describes the angle ∠A₁o′A₂, the third one is thedistance between spherical joint center A₃ and the point o′, the fourthand the fifth are the limb length expressions of limb 1 and limb 3 whichare the same with (1).

Substituting (2) into (3) gives

$\begin{matrix}\{ \begin{matrix}{{f_{1}( {\alpha_{2},\alpha_{4},1} )} = 0} \\{{f_{2}( {\alpha_{2},\alpha_{4},1,x_{1},y_{1},z_{1}} )} = 0} \\{{{f_{3}( {\alpha_{2},\alpha_{4},1,x_{1},y_{1},z_{1}} )} + x_{1}^{2} + y_{1}^{2} + z_{1}^{2}} = 0} \\{{{f_{4}( {1,x_{1},y_{1},z_{1}} )} + x_{1}^{2} + y_{1}^{2} + z_{1}^{2}} = 0} \\{{{f_{5}( {\alpha_{2},\alpha_{4},1,x_{1},y_{1},z_{1}} )} + {k_{1}^{2}( {x_{1}^{2} + y_{1}^{2} + z_{1}^{2}} )}} = 0}\end{matrix}  & (4)\end{matrix}$

where f_(i)(•) is a linear function of (x₁, y₁, z₁)and include cosineand sine functions of the angle a_(i).

The last three equations in (4) are linear functions of (x₁ ², y₁ ², z₁²). Then two new equations can be obtained from these three byeliminating (x₁ ², y₁ ², z₁ ²) as

$\begin{matrix}\{ \begin{matrix}{{f_{6}( {\alpha_{2},\alpha_{4},1,x_{1},y_{1},z_{1}} )} = 0} \\{{f_{7}( {\alpha_{2},\alpha_{4},1,x_{1},y_{1},z_{1}} )} = 0}\end{matrix}  & (5)\end{matrix}$

Thus, (x₁, y₁, z₁) can be eliminated from f₂ in (4) and f₆, f₇ in (5).Substituting the results into f₁, f₄ and replacing cos a_(i)=(1−t_(i)²)/(1+t_(i) ²), sin a_(i)=2t_(i)/(1+t_(i) ²), gives

$\begin{matrix}\{ {\begin{matrix}{{f_{8}( {1,t_{2}^{2},t_{4}^{2},{t_{2}t_{4}},{t_{2}^{2}t_{4}^{2}}} )} = 0} \\{{f_{9}( {1,t_{2}^{2},t_{2}^{4},t_{2}^{6},t_{2}^{8},{t_{2}^{i}t_{4}^{j}},\ldots \mspace{14mu},{t_{2}^{8}t_{4}^{8}}} )} = 0}\end{matrix}( {j \neq 0} )}  & (6)\end{matrix}$

Following Sylvester's dialytic elimination method for the two equationsin (6), a univariate equation in t₄ of degree 32 is obtained:

$\begin{matrix}{{\sum\limits_{i = 0}^{+ 16}{h_{i}t_{4}^{2i}}} = 0} & (7)\end{matrix}$

where coefficient h_(i) are real constants depending on constantmechanism parameters and input data only.

Solving (7), 32 solutions for t₄ can be obtained. Then, t₂ can be solvedby substituting each solution of t₄ back to the equations in (6) andsolving the common roots. Following this, (x₁, y₁, z₁) can be linearlysolved by substituting each pair of solutions of t₂ and t₄ into f₂ in(4) and f₆, f₇ in (5). Based on this, 32 sets of solutions of t₂, t₄ and(x₁, y₁, z₁) are obtained and the spherical joint center A_(i) can becalculated by substituting a_(i)=2Arc Tan(t_(i)) into (2). Then, theplatform position and orientation can be determined using the threespherical joint centers as:

$\begin{matrix}\{ \begin{matrix}{y^{\prime} = {( {a_{2o} - a_{4o}} )/{{a_{2o} - a_{4o}}}}} \\{z^{\prime} = {( {a_{3o} - {( {a_{2o} + a_{4o}} )/2}} ) \cdot {y^{\prime}/( {a_{1}\cos \; ( \varphi_{a\; 1} )} )}}} \\{{x^{\prime} = {y^{\prime} \times z^{\prime}}},{R = ( {x^{\prime},y^{\prime},z^{\prime}} )},{p = {( {a_{2o} + a_{4o}} )/2}}}\end{matrix}  & (8)\end{matrix}$

4. Jacobian Matrices and Singularity Loci for Maximum Singularity-FreeWorkspace:

4.1 Screw Theory Based Overall Jacobian Matrix:

The infinitesimal twist of the moving platform of the 2RPS-2UPS parallelmechanism can be written as a linear combination of instantaneous twistsof each limb:

$\begin{matrix}\{ \begin{matrix}{S_{p} = {{{\overset{.}{\varphi}}_{i\; 1}S_{i\; 1}} + {{\overset{.}{\varphi}}_{i\; 2}S_{i\; 2}} + {{\overset{.}{l}}_{i\;}S_{i\; 3}} + {{\overset{.}{l}}_{i}S_{i\; 4}} + {{\overset{.}{\varphi}}_{i\; 5}S_{i\; 5}} + {{\overset{.}{\varphi}}_{i\; 6}S_{i\; 6}}}} & ( {{i = 1},3} ) \\{S_{p} = {{{\overset{.}{\varphi}}_{i\; 1}S_{i\; 1}} + {{\overset{.}{l}}_{i}S_{i,3}} + {{\overset{.}{\varphi}}_{i\; 4}S_{i\; 4}} + {{\overset{.}{\varphi}}_{i\; 5}S_{i\; 5}} + {{\overset{.}{\varphi}}_{i\; 6}S_{i\; 6}}}} & ( {{i = 2},4} )\end{matrix}  & (9)\end{matrix}$

where S_(p) represents the infinitesimal twist of the moving platform,S_(ij) (j=1,2,3,4,5,6) denotes the unit screw of the jth 1-DOF joint inlimb i, {dot over (l)}_(i) is the distance rate of the prismatic jointin limb i, and {dot over (φ)}_(ij) (j=1,2,4,5,6) represent angular ratesof the universal joint and spherical joint in limb i.

Thus by locking the active joints in the limbs temporarily and takingthe reciprocal product on both sides of (9), four actuation reciprocalscrews and two geometric constraint screws can be found to give thefollowing expression:

$\begin{matrix}{{\begin{bmatrix}S_{11}^{r} \\S_{21}^{r} \\S_{31}^{r} \\S_{41}^{r} \\S_{22}^{r} \\S_{42\;}^{r}\end{bmatrix} \circ S_{p}} = {{\begin{bmatrix}u_{1} & {{Ra}_{1} \times u_{1}} \\u_{2} & {{Ra}_{2} \times u_{2}} \\u_{3} & {{Ra}_{3} \times u_{3}} \\u_{4} & {{Ra}_{4} \times u_{4}} \\x & {{Ra}_{2} \times x} \\x & {{Ra}_{4} \times x}\end{bmatrix} \circ S_{p}} = {{J \circ S_{p}} = {{\begin{bmatrix}J_{a} \\J_{c}\end{bmatrix} \circ S_{p}} = {\begin{bmatrix}{\overset{.}{l}}_{1} \\{\overset{.}{l}}_{2} \\{\overset{.}{l}}_{3} \\{\overset{.}{l}}_{4} \\0 \\0\end{bmatrix} = \begin{bmatrix}{\overset{.}{l}}_{a} \\0\end{bmatrix}}}}}} & (10)\end{matrix}$

where x=(1,0,0)^(T), u_(i) is the unit vector of the i^(th) limbdirection, {dot over (l)}_(a) represents a vector of the four linearinput rates, S_(i1) ^(r) (i=1,2,3,4) is the actuation screw reciprocalto all joint motion screws in the ith limb except the prismatic jointscrew S_(i3) and it is collinear with the limb, S_(i2) ^(r) (i=2,4) isthe reciprocal screw of geometric constraint to all motion screws inlimb i and it passes through the spherical joint center with thedirection parallel to the revolute joint.

Thus J is the 6 by 6 overall Jacobian matrix. The first four rows arethe four actuation forces represented by actuation Jacobian J_(a) in(10) while the last two rows are constraint forces denoted by constraintJacobian J_(c). The zero determinant of the overall Jacobian Jrepresents singular velocity mappings and singular configurations of theparallel mechanism. Due to the some mechanism arrangement symmetry, likethe limb 2 and limb 4, and the design that all joints are on the sameplane for both the base and platform, the singularity equation from thedeterminant of the Jacobian J is simplified and further study can alsoconsider the method in to have the potential to simplify the finalequation.

4.2 Dimensionless Jacobian Matrix for Kinematics Performance Evaluation:

Since the 2RPS-2UPS parallel mechanism has two translational and tworotational motions, the actuation Jacobian J_(a) involves both linearand angular velocity mappings. Thus, its singular values are not in thesame unit and its condition number cannot be used directly forkinematics performance evaluation. Following this, a dimensionlessJacobian matrix is introduced. One approach is to map the platformvelocity to linear velocities in some directions at selected points onthe platform representing the platform mobility. This mapping provides auniform unit between the linear platform point velocities and linearactuation limb inputs. Considering the motion type of the 2RPS-2UPSparallel mechanism, four linear velocities at three points on theplatform are selected, FIG. 4.

To present the two translational motion of the platform along y-axis andz-axis, linear velocities along n₁=(0,1,0)^(T) and n₂=(0,0,1)^(T) atpoint O′ are selected. For the two rotation motions about x-axis andy-axis, linear velocities along n₃=(0,1,0)^(T) at point P₃ and alongn₄=(0,0,1)^(T) at point P₄ are selected. Then these linear velocitiescan be expressed by the platform velocity in the platform coordinateframe as:

v_(p)=[v₁ v₂ v₃ v₄]^(T)=J_(p) M^(T) S_(p)   11

where v_(i) is the linear velocity along n_(i) at the selected point,

${J_{p} = \begin{bmatrix}S_{n\; 1} & S_{n\; 2} & S_{n\; 3} & S_{n\; 4}\end{bmatrix}^{T}},{M = \begin{bmatrix}R & 0 \\0 & R\end{bmatrix}},$

S_(ni)=[n_(i) p_(i)×n_(i)]^(T), (i=1,2,3,4), p_(i) is the vector ofpoint i at which linear velocities are selected and p₁=p₂=(0,0,0)^(T),p₃=(0,0,a₁)^(T), p₄=(−a₁,0,0)^(T).

From (10), there is

S _(p)=(J ^(T) J)⁻¹ J _(a) ^(T) {dot over (l)} _(a)   12

Combining (11) and (12), the selected linear velocities can be obtaineddirectly from the linear actuation input velocities:

v _(p) =J _(p) M ^(T) (J ^(T) J)⁻¹ J _(a) ^(T) {dot over (l)} _(a) =J_(D) ⁻¹ {dot over (l)} _(a)   13

where J_(D)=(J_(p) M^(T) (J^(T) J)⁻¹ J_(a) ^(T))⁻¹ is the 4×4dimensionless Jacobian matrix.

4.3 Parameterization and Singularity Loci:

From section 4.1, the overall Jacobian matrix J maps the velocitiesbetween the manipulator and the actuation input while satisfying thegeometric constraints. Once the manipulator meets the singularconfiguration, this mapping loses its function and the rank of theJacobian matrix decreases to less than 6. This can be also interpretedthat the four actuation forces and two constraint forces in J arelinearly dependent. Inversely, identifying the dependent conditions forthe constraint forces in the workspace will reveal the singularconfigurations of the manipulator. This can be analyzed by taking thedeterminant of J to be zero.

In order to illustrate the singularity loci in a uniform unit, themotion of the platform is described by two rotation angles (α, β) in therotation matrix R and a lengths d and an angle θ in the translationvector p=(0,p_(y),p_(z))^(T)=d(0, cos(θ), sin(θ))^(T). For differentlength d, 3D singularity loci in the coordinates (α, β, θ) can be shown,FIG. 5.

In FIG. 5(a), the singularity loci correspond to the mechanismconfiguration with two UPS limbs located on the two sides of the planeB₂A₂A₄B₄ formed by the two RPS limbs. One singular configuration isshown in FIG. 5(a) which is a Type 5a singularity as the six skewconstraint forces (red lines) in the overall Jacobian matrix form a5-system with one redundant. Similarly, another example is shown in FIG.5(b) in which the singularity loci are for the configuration with thetwo UPS limbs on the same side of the plane B₂A₂A₄B₄. This singularconfiguration is also a Type 5a singularity.

4.4 Maximum Singularity-Free Workspace:

Following the singularity loci in section 4.3, the maximum singularityfree workspace is defined as the maximum workspace starting from theinitial configuration (α=0, β=0, θ=π/2, variable d) to the first pointmeeting the singularity loci. An example with a given d is shown in FIG.6(a) in which the red part is the singularity loci and the light bluecircles represent the maximum singularity-free workspace with differentθ. One of the circles is illustrated in FIG. 6(b) in which the blackcircle has the maximum radius on the (α, β) plane starting from (α=0,β=0) to the singularity loci in blue. The integration of these circlesin the (α, β, θ) coordinate gives the maximum singularity-free workspacecorresponding to a given d and mechanism geometric constraints includingpassive joint ranges. There are several ways to find the maximum circleas shown in and it's not repeated here. Then by integrating all possibleplatform translation vector length d, the maximum singularity-freeworkspace of the 2RPS-2UPS parallel mechanism is obtained.

5. Maximum-Singularity-Free Workspace and Kinematics Performance BasedOptimal Design:

5.1 Design Variables and Performance Indices:

As discussed in Section 2.1 and FIG. 3, key parameters of the 2RPS-2UPSparallel mechanism in the optimization are the base and platform sizes(b and a) defined by the joint distance between limb 2 and limb 4, andthe location parameters of the two UPS limbs with (φ_(b1), φ_(a1),φ_(b3), φ_(a3), a₁=a₃, b₁=b₃). To have a relative relation, the lengthparameters are normalized by the base size b as λ_(a)=a/b, λ_(a1)=a₁/b,λ_(b1)=b₁/b, λ_(d)=d/b, and a₃, b₃ are replaced by a₁, b₁. Thus, λ_(a)represents the ratio between the platform and base sizes, λ_(a1) andλ_(b1) show the ratios of the spherical and universal joint locationradii over the base size, λ_(d) represents the translation of theplatform. Based on the singularity loci analysis in section 3.3, it isfound that the mechanism has a Type 5d singularity as in FIG. 5(a) whenthe two UPS limbs are separated by the plane B₂A₂A₄B₄ formed by the twoRPS limbs. Thus, in the following, the two UPS limbs are placed on oneside. Considering the symmetrical property of two sides with respect tothe plane B₂A₂A₄B₄, a range of (π, 3π/2) corresponding to the right sideof the plane B₂A₂A₄B₄ for all the four location angles (φ_(b1), φ_(a1),φ_(b3), φ_(a3)) will be used. In addition to this, mechanicalconstraints including maximum passive joint angles and limb interferenceshould also be considered in the optimization. In the following, passivejoint angles are limited in the range as −ψ_(max)≦ψ_(i)≦ψ_(max), whereψ_(i) denotes rotation angle from its home position of any revolutejoint, spherical joint and universal joint while ψ_(max) is given π/4.The minimum distance between any two limbs is limited to be 0.01 toavoid limb interference and the limb lengths are determined by limitingthe platform translation with 0.8≦λ_(d)≦1.4.

For the kinematics performance, condition number k_(i)=σ_(max)/σ_(min),(σ_(max) and σ_(min) are the maximum and minimum singular values of thedimensionless Jacobian J_(D)) is a widely used parameter in parallelmechanism design and optimization. As mentioned above, the conditionnumber is calculated using the dimensionless Jacobian considering thecoupled mapping with linear and angular velocities.

The optimal design of the 2RPS-2UPS parallel mechanism in this paper isto find the best parameter set to have maximum singularity-freeworkspace with good kinematics performance. Thus, the optimal designcost function can be given as:

$\begin{matrix}\{ {\begin{matrix}\max & V \\\max & k\end{matrix},{k = \frac{V}{\int_{V}{k_{i}{V}}}}}  & 14\end{matrix}$

where V is the maximum singularity-free workspace, k is the inverseaverage condition number in the workspace V and is between 0 and 1. Thebest kinematics performance corresponds to the value 1 when the velocitymapping is isotropic.

5.2 Optimal Design:

5.2.1 Effect of Parameters a₁ and b₁

Based on the above analysis, the maximum singularity-free workspace andkinematics performance are calculated with variable λ_(a), λ_(a1), andλ_(b1) as shown in FIG. 7 in which the location angles are given as(φ_(b1)=φ_(a1)=3π/4, φ_(b3)=φ_(a3)=5π/4). In FIG. 7, the solid lines arefor λ_(a)=0.5 and dashed lines are with λ_(a)=0.8 which is the same forthe following analysis. For each λ_(a) (solid or dashed) case, thedifferent color lines are for variable λ_(b1) and in each line λ_(a1)changes in the range (0.4, 1.6) as shown in the horizontal axis. In FIG.7, comparing the solid and dashed lines it can be seen that a smallerratio between the platform size and the base size generally gives largermaximum singularity-free workspace V and better kinematics performance.Similarly, the increase of ratio λ_(b1) (the location radius of the UPSlimbs on the base over the base size) decreases workspace V and thekinematics performance. In FIG. 7(a), when λ_(b1)=0.6 is a separate line(green lines) at which the workspace is balanced at a value even λ_(a1)changes. The workspace increases with the increase of λ_(a1) whenλ_(b1)>0.6 (blue, black, purple) and it decreases when λ_(b1)<0.6 (red).For the kinematics performance in FIG. 7(b), a bigger number of λ_(a1)gives better performance based on the inverse average condition number.In both figures in FIG. 7, there are some points close to zero. This isdue to the singularity loci being close to the initial configuration(α=0, β=0, θ=π/2, variable d) resulting in a very small singularity-freeworkspace and poor kinematics performance.

5.2.2 Effect of Parameters φ_(b1) and φ_(a1)

Parameters φ_(b1) and φ_(a1) represent the locations of the two ends oflimb 1 on the platform and on the base. From FIG. 7 and the analysis,λ_(b1)=0.6 and λ_(a1)=0.5 is selected in the following location angleoptimization. The effect on both singularity-free workspace V andkinematics performance of parameter φ_(b1) and φ_(a1) whileφ_(b3)=φ_(a3)=5π/4 is shown in FIG. 8 in which solid lines are forλ_(a)=0.5 and dashed lines for λ_(a)=0.8. The different color lines arefor different angles of φ_(a1), representing the location of thespherical joint in limb 1 on the platform. Similar with FIG. 7, asmaller ratio λ_(a) of platform over base size provides a largersingularity-free workspace and better performance by comparing the samecolor lines in solid and dashed forms. Comparing among the colors inFIG. 8(a), when φ_(a1) is smaller, e.g. when the spherical joint of limb1 on the platform is close to that of limb 2, the singularity-freeworkspace is larger. For a given φ_(a1), V increases with the increaseof φ_(b1) and the spherical joint center of limb 1 is close to that oflimb 3 on the platform. When φ_(b1) increases further, the workspacedecreases. This is a little different for the kinematics performance inFIG. 8(b) in which both smaller φ_(a1) (red line) and bigger φ_(a1)(purple line) show bigger values of the inverse condition number. Thisindicates that the spherical joint of limb 1 should be close to eitherlimb 2 or limb 4. To have better kinematics performance for a givenφ_(a1), φ_(b1) should be big when φ_(a1) is small and φ_(b1) should besmall when φ_(a1) is big. Thus the spherical joint and the universaljoint in limb 1 should be away to each other.

It is noted that parameters φ_(b3) and φ_(a3) have the same effect asφ_(a1) and φ_(b1), considering the symmetrical structure of the parallelmechanism and same form of limb 1 and limb 3. Thus, the above resultsfrom FIG. 8 can be directly applied to limb 3.

5.2.3 Effect of Parameters φ_(a1) and φ_(a3)

The two parameters φ_(a1) and φ_(a3) represent the locations of thespherical joints of limb 1 and limb 3 on the platform. FIG. 9 showstheir effect on the maximum singularity-free workspace and kinematicsperformance while φ_(b1)=3π/4, φ_(b3)=5π/4. Solid lines are forλ_(a)=0.5, dashed lines for λ_(a)=0.8 and different color lines fordifferent angles of φ_(a1). It can be seen that a smaller λ_(a) givesmuch better kinematics performance and a larger singularity-freeworkspace when φ_(a3) is large, e.g. the spherical joint of limb 3 isclose to that of limb 4 on the platform. For V, it is preferable to havesmaller φ_(a1) and bigger φ_(a3) while for the kinematics performance itis better to have both of them close to π as in FIG. 9.

5.2.4 Effect of Parameters φ_(b1) and φ_(b3)

The two parameters φ_(b1) and φ_(b3) represent the locations of theuniversal joints of limb 1 and limb 3 on the base. The result is shownin FIG. 10 in which solid lines are for λ_(a)=0.5 and dashed lines forλ_(a)=0.8 while different colors are for different φ_(b1) increasingfrom red, green, blue, black to purple. In general, dashed lines arelower than solid ones, indicating that a smaller λ_(a) is better. Forthe maximum singularity-free workspace in FIG. 10(a), smaller φ_(b1)corresponds to larger V when φ_(b3) is less than π, while it is oppositewhen φ_(b3) is less than 5π/4. This means that it's better to have thetwo universal joints of limb 1 and limb 3 to be close to have a largersingularity-free workspace. The trend of the kinematics performancecurves is much clearer as in FIG. 10(b) comparing with the workspaceones. It is seen that a smaller φ_(b3) and a larger φ_(b1) give the bestkinematics performance as shown by the top purple line. This needs thetwo UPS limbs to be crossed and the two universal joints far away fromeach other.

To conclude the above analysis, the ratio (a/b) between the platformsize (a) and the base size (b) should be small for both singularity-freeworkspace and kinematics performance. The spherical joints on theplatform (a₁) and universal joints on the base (b₁) in limb 1 and limb 3should be close to limb 2 and limb 4 to have large singularity-freeworkspace while they have to be crossed to give good kinematicsperformance. To compromise these and avoid limb interference, a “V”shape assembly of limb 1 and limb 3 can be obtained. For this, there arealso two solutions while one is to assemble the two spherical joints onthe platform close to each other and the other is to let the twouniversal joints on the base close to each other. However, comparingFIG. 9 and FIG. 10 it is clear that a better combination of bothsingularity-free workspace and kinematics performance can be obtained byselecting the latter way with the two universal joints close to eachother and the two spherical joints separate on the platform. Based onthese, the following optimized mechanism design is obtained, FIG. 11.

6. An Optimized 2T2R Parallel Mechanism for AFP

In this section a design example is given by following the aboveoptimization procedures. The previous section gives the whole map formechanism design parameters for singularity-free workspace andkinematics performance. In specific design, effect from selectedmechanical components on the maximum singularity-free workspace shouldbe also figured out.

In general, the platform and base sizes (a and b) can be determined byconsidering the actual application requirement. To hold the fiberplacement head, a minimum size of the platform should be used. Followingthe above analysis, this minimum size should be selected to have a smallplatform size over base size ratio (a/b) for large singularity-freeworkspace and good kinematics performance. The selection for thelocation of the limb 1 and limb 3 on the platform (a₁) and on the base(b₁) follows the rule shown in FIG. 6 in which a relatively small b₁ anda compromised a₁ should be selected. Limb lengths and strokes andmaximum passive joint angles relate to component selection which can beoptimized by comparing different sizes. Here the limb lengths andstrokes are for the prismatic joints and the maximum passive jointangles are for the spherical joints, universal joints and revolutejoints. The former can be represented by a stroke ratio as

$\begin{matrix}{\lambda_{l} = \frac{l_{{ma}\; x} - l_{m\; i\; n}}{l_{m\; i\; n}}} & 15\end{matrix}$

where l_(max) and l_(min) represent the maximum and minimum limb lengthsrespectively while all the limbs have the same size. In the following anexample will be given to optimize the selection of λ_(l) and ψ_(max)based on the optimized mechanism configuration in FIG. 11. Basicparameters are given as a=0.18, b=0.3, a₁=0.18, b₁=0.18, φ_(b1)=17π/18,φ_(a1)=3π/4, φ_(b3)=19π/18, φ_(a3)=5π/4. The workspace volume andkinematics performance are calculated as in FIG. 12 for which limbinterference is included with minimum distance between any two limbs as0.01.

From FIG. 12, it can be seen that the singularity-free workspaceincreases with the increase of the stroke ratio λ_(l) but the kinematicsperformance decreases at the same time. This is due to the expandedworkspace part closer to singularity configurations and hence thekinematics performance is worse. Similarly, a larger maximum passiverotation joint angle, ψ_(max), will give a larger singularity-freeworkspace but worse average kinematics performance as in FIG. 12. Theseconclusions are actually intuitive since a larger joint workspace willallow a larger platform workspace. FIG. 12 not only shows this effect,but also gives a way in selecting the limb stroke and passive rotationjoints. To have the singularity-free workspace V=4 as an example, theavailable stroke ratio λ_(l) is between 0.54 and 0.75 giving the maximumpassive rotation joint angle ψ_(max) between 5π/36 and π/4. Once astroke is selected, for example λ_(l)=0.6, the corresponding kinematicsperformance can be obtained in FIG. 12(b) with 0.127≦k≦0.133. It's alsonoted that to have a better kinematics performance, the smallestψ_(max)=5π/36 should be selected in the design corresponding to the redline.

The foregoing descriptions of specific embodiments of the presentinvention have been presented for purposes of illustration anddescription. They are not intended to be exhaustive or to limit thepresent invention to the precise forms disclosed, and obviously manymodifications and variations are possible in light of the aboveteaching. The embodiments were chosen and described in order to bestexplain the principles of the present invention and its practicalapplication, and to thereby enable others skilled in the art to bestutilize the present invention and various embodiments with variousmodifications as are suited to the particular use contemplated. It isunderstood that various omissions and substitutions of equivalents arecontemplated as circumstances may suggest or render expedient, but suchomissions and substitutions are intended to cover the application orimplementation without departing from the spirit or scope of the presentinvention.

1. A robot apparatus comprising: a parallel mechanism comprising twoRevolute-Prismatic-Spherical joint (RPS) limbs and twoUniversal-Prismatic-Spherical joint (UPS) limbs.
 2. The robot apparatusas claimed in claim 1 wherein said parallel mechanism is adapted to beused for composites manufacturing.
 3. The robot apparatus as claimed inclaim 2 wherein said parallel mechanism is adapted to operate anAutomated Fiber Placement Head (AFP).
 4. The robot apparatus as claimedin claim 1 wherein the parallel mechanism provides a 4 Degree-Of-Freedom(DOF) movement comprising two rotations and two translations.
 5. Therobot apparatus as claimed in claim 1 wherein the prismatic joints areadapted to provide linear motion with enough power.
 6. The robotapparatus as claimed in claim 5 wherein the prismatic joints areelectric, hydraulic or pneumatic.
 7. The robot apparatus as claimed inclaim 1 wherein the spherical joints are ball joint or seriallyconnected revolute joints.
 8. The robot apparatus as claimed in claim 1wherein the universal joints are cross-link connected or seriallyconnected revolute joints.
 9. The robot as claimed in claim 1 furthercomprising: a platform; and a base; wherein the revolute and universaljoints are adapted to be connected to the base, the spherical joints areadapted to be connected to the platform and the prismatic joints areadapted to be connected intermediate the revolute/universal and thespherical joints.
 10. The robot apparatus as claimed in claim 9 whereinthe two revolute joints in the RPS limbs are located parallel to eachother on to the base and adapted to make the two limbs work in the sameplane perpendicular to the revolute joints.
 11. The robot apparatus asclaimed in claim 10 wherein the joints are in 3D space and notconstrained on the same plane.
 12. The robot apparatus as claimed inclaim 9 wherein the base is movable enabling a 6 degree-of-freedommovement for the parallel mechanism.
 13. The robot apparatus as claimedin claim 9 further comprising an Automated Fiber Placement Head (AFP)adapted to be connected to the base.
 14. The robot apparatus as claimedin claim 10 adapted for use in composites manufacturing.
 15. The robotapparatus as claimed in claim 14 adapted to reinforce fibers and matrixmaterials comprising thermoset and thermoplastic tapes or prepregs anddry carbon fiber unidirectional materials.
 16. The robot apparatus asclaimed in claim 1 adapted for use in camera orientation, materialgrasping and manipulation, machining tool, laser cutting, 3D printing,stabilization platform, motion simulator, automatic painting, automaticwelding or non destructive testing.